Theorem lcfrlem42

Description: Lemma for lcfr . Eliminate nonzero condition. (Contributed by NM, 11-Mar-2015)

	Ref	Expression
Hypotheses	lcfrlem38.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcfrlem38.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	lcfrlem38.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcfrlem38.p	⊢ + = ( +g ‘ 𝑈 )
	lcfrlem38.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	lcfrlem38.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	lcfrlem38.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
	lcfrlem38.q	⊢ 𝑄 = ( LSubSp ‘ 𝐷 )
	lcfrlem38.c	⊢ 𝐶 = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
	lcfrlem38.e	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘ ( 𝐿 ‘ 𝑔 ) )
	lcfrlem38.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	lcfrlem38.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑄 )
	lcfrlem38.gs	⊢ ( 𝜑 → 𝐺 ⊆ 𝐶 )
	lcfrlem38.xe	⊢ ( 𝜑 → 𝑋 ∈ 𝐸 )
	lcfrlem38.ye	⊢ ( 𝜑 → 𝑌 ∈ 𝐸 )
Assertion	lcfrlem42	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐸 )

Proof

Step	Hyp	Ref	Expression
1		lcfrlem38.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcfrlem38.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		lcfrlem38.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		lcfrlem38.p	⊢ + = ( +g ‘ 𝑈 )
5		lcfrlem38.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
6		lcfrlem38.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
7		lcfrlem38.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
8		lcfrlem38.q	⊢ 𝑄 = ( LSubSp ‘ 𝐷 )
9		lcfrlem38.c	⊢ 𝐶 = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
10		lcfrlem38.e	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘ ( 𝐿 ‘ 𝑔 ) )
11		lcfrlem38.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12		lcfrlem38.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑄 )
13		lcfrlem38.gs	⊢ ( 𝜑 → 𝐺 ⊆ 𝐶 )
14		lcfrlem38.xe	⊢ ( 𝜑 → 𝑋 ∈ 𝐸 )
15		lcfrlem38.ye	⊢ ( 𝜑 → 𝑌 ∈ 𝐸 )
16	1 3 11	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
17		eqid	⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
18	1 2 3 17 6 7 8 10 11 12 14	lcfrlem4	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑈 ) )
19	1 2 3 17 6 7 8 10 11 12 15	lcfrlem4	⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝑈 ) )
20	17 4	lmodcom	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ ( Base ‘ 𝑈 ) ∧ 𝑌 ∈ ( Base ‘ 𝑈 ) ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )
21	16 18 19 20	syl3anc	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )
22	21	adantr	⊢ ( ( 𝜑 ∧ 𝑋 = ( 0g ‘ 𝑈 ) ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )
23	11	adantr	⊢ ( ( 𝜑 ∧ 𝑋 = ( 0g ‘ 𝑈 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
24	12	adantr	⊢ ( ( 𝜑 ∧ 𝑋 = ( 0g ‘ 𝑈 ) ) → 𝐺 ∈ 𝑄 )
25	15	adantr	⊢ ( ( 𝜑 ∧ 𝑋 = ( 0g ‘ 𝑈 ) ) → 𝑌 ∈ 𝐸 )
26		eqid	⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 )
27		simpr	⊢ ( ( 𝜑 ∧ 𝑋 = ( 0g ‘ 𝑈 ) ) → 𝑋 = ( 0g ‘ 𝑈 ) )
28	1 2 3 4 6 7 8 23 24 10 25 26 27	lcfrlem7	⊢ ( ( 𝜑 ∧ 𝑋 = ( 0g ‘ 𝑈 ) ) → ( 𝑌 + 𝑋 ) ∈ 𝐸 )
29	22 28	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑋 = ( 0g ‘ 𝑈 ) ) → ( 𝑋 + 𝑌 ) ∈ 𝐸 )
30	11	adantr	⊢ ( ( 𝜑 ∧ 𝑌 = ( 0g ‘ 𝑈 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
31	12	adantr	⊢ ( ( 𝜑 ∧ 𝑌 = ( 0g ‘ 𝑈 ) ) → 𝐺 ∈ 𝑄 )
32	14	adantr	⊢ ( ( 𝜑 ∧ 𝑌 = ( 0g ‘ 𝑈 ) ) → 𝑋 ∈ 𝐸 )
33		simpr	⊢ ( ( 𝜑 ∧ 𝑌 = ( 0g ‘ 𝑈 ) ) → 𝑌 = ( 0g ‘ 𝑈 ) )
34	1 2 3 4 6 7 8 30 31 10 32 26 33	lcfrlem7	⊢ ( ( 𝜑 ∧ 𝑌 = ( 0g ‘ 𝑈 ) ) → ( 𝑋 + 𝑌 ) ∈ 𝐸 )
35	11	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ ( 0g ‘ 𝑈 ) ∧ 𝑌 ≠ ( 0g ‘ 𝑈 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
36	12	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ ( 0g ‘ 𝑈 ) ∧ 𝑌 ≠ ( 0g ‘ 𝑈 ) ) ) → 𝐺 ∈ 𝑄 )
37	13	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ ( 0g ‘ 𝑈 ) ∧ 𝑌 ≠ ( 0g ‘ 𝑈 ) ) ) → 𝐺 ⊆ 𝐶 )
38	14	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ ( 0g ‘ 𝑈 ) ∧ 𝑌 ≠ ( 0g ‘ 𝑈 ) ) ) → 𝑋 ∈ 𝐸 )
39	15	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ ( 0g ‘ 𝑈 ) ∧ 𝑌 ≠ ( 0g ‘ 𝑈 ) ) ) → 𝑌 ∈ 𝐸 )
40		simprl	⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ ( 0g ‘ 𝑈 ) ∧ 𝑌 ≠ ( 0g ‘ 𝑈 ) ) ) → 𝑋 ≠ ( 0g ‘ 𝑈 ) )
41		simprr	⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ ( 0g ‘ 𝑈 ) ∧ 𝑌 ≠ ( 0g ‘ 𝑈 ) ) ) → 𝑌 ≠ ( 0g ‘ 𝑈 ) )
42	1 2 3 4 5 6 7 8 9 10 35 36 37 38 39 26 40 41	lcfrlem41	⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ ( 0g ‘ 𝑈 ) ∧ 𝑌 ≠ ( 0g ‘ 𝑈 ) ) ) → ( 𝑋 + 𝑌 ) ∈ 𝐸 )
43	29 34 42	pm2.61da2ne	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐸 )